<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">p</font> be  <a href="goboard1.html#V1" title="GOBOARD1:attr.1">increasing</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c2">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:63"/><i><font color="Green" title="E44">A1</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font>
 ;<br/>

<a NAME="E2:63"/>
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Olive" title="b2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b2">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">i</font>, <font color="Maroon" title="c4">j</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:63_1"/><i><font color="Green" title="E45">A2</font></i>: 
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c2">n</font></span>)</span>
 <b>and </b><br/><a NAME="E2:63_1"/><i><font color="Green" title="E46">A3</font></i>: 
<font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c2">n</font></span>)</span>
 <b>and </b><br/><a NAME="E3:63_1"/><i><font color="Green" title="E47">A4</font></i>: 
<font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">j</font>
 ;<br/>

<a NAME="E4:63_1"/><i><font color="Green" title="E48">A5</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c2">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:63"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="integra1.html#E1:63_1"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></i>;<br/>
<a NAME="E5:63_1"/><i><font color="Green" title="E49">A6</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c2">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:63"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="integra1.html#E2:63_1"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></i>;<br/>
<a NAME="E6:63_1"/>
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:63_1"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:63_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c2">n</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E8:63_1"/><b>then </b><i><font color="Green" title="E50">A7</font></i>: 
( 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">n</font> )
 
<b>by </b><i><a class="txt" href="integra1.html#E1:63"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></i>;<br/>
<a NAME="E9:63_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E10:63_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font> )
 
<b>by </b><i><a class="txt" href="integra1.html#E8:63_1"><i><font color="Green" title="E50">A7</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/>
<a NAME="E11:63_1"/><b>then </b>
<font color="Maroon" title="c3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E12:63_1"/><b>then </b><i><font color="Green" title="E51">A8</font></i>: 
<font color="Maroon" title="c3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E13:63_1"/>
<font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E2:63_1"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E14:63_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">j</font> &amp; <font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c2">n</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E15:63_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">j</font> &amp; <font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">n</font> )
 
<b>by </b><i><a class="txt" href="integra1.html#E1:63"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></i>;<br/>
<a NAME="E16:63_1"/><b>then </b><i><font color="Green" title="E52">A9</font></i>: 
( 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c4">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/>
<a NAME="E17:63_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E18:63_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c4">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font> )
 
<b>by </b><i><a class="txt" href="integra1.html#E16:63_1"><i><font color="Green" title="E52">A9</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E19:63_1"/><b>then </b>
<font color="Maroon" title="c4">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E20:63_1"/><b>then </b><i><font color="Green" title="E53">A10</font></i>: 
<font color="Maroon" title="c4">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E21:63_1"/>
<font color="Maroon" title="c3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="txt" href="integra1.html#E3:63_1"><i><font color="Green" title="E47">A4</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<b>hence </b><a NAME="E22:63_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c4">j</font>
 <b>by </b><i><a class="txt" href="integra1.html#E4:63_1"><i><font color="Green" title="E48">A5</font></i></a>, <a class="txt" href="integra1.html#E5:63_1"><i><font color="Green" title="E49">A6</font></i></a>, <a class="txt" href="integra1.html#E12:63_1"><i><font color="Green" title="E51">A8</font></i></a>, <a class="txt" href="integra1.html#E20:63_1"><i><font color="Green" title="E53">A10</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:63"/>
<font color="Maroon" title="c1">p</font> <a href="rfinseq.html#K1" title="RFINSEQ:func.1">/^</a> <font color="Maroon" title="c2">n</font> is  <a href="goboard1.html#V1" title="GOBOARD1:attr.1">increasing</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <b>by </b><i><a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/>


</div>
